FIGURE EX3.19 shows vectors A and B. What is C = A + B? Write your answer in component form using unit vectors

Video transcript

Hey, everyone in this problem, we're asked to use vectors M and N given in the figure below to determine P which is equal to M plus N and to give the result in component form with unit vector. So in our figure here we have vector M which is drawn in blue, it has a length of 6 m. It's pointing up from the origin into quad one making an angle of degrees with the vertical. Then we have vector N drawn in red or orange. Here has a length of 5 m again pointing from the origin. But down into quadrant four making an angle of 40 degrees. With the, we have four answer choices A through D, all of them have the vector written in component form K with an I component and AJ component. They just differ in which value they have for each of the I and J component. And we're gonna come back to those values as we work through this problem. Now, we're given two vectors, we wanna find the sum of the two. Let's recall that if we're adding two vectors, we can add them component one. OK. So we can break M and N up into the X and Y or I and J component and then add the corresponding component. So let's start with M and we're gonna break this down. We're gonna draw a triangle above K because of where the angle is that we're given, we're gonna break this down into the XX component, which is this horizontal component and the Y component, the vertical component, we have MX and my respect. Now, if we wanna write MX, OK. Well, this is gonna be related to our angle through sign because we're talking about the opposite side. So MX is gonna be equal to M in the magnitude of the hypo multiplied by ST of 30 degrees. OK. So this is gonna be 6 m multiplied by sine of 30 degrees. And we're gonna leave that as a sign of 30 degrees. We're gonna simplify at the very end in one step so that we can avoid some round off error there. We're gonna do the same for the vertical component. My is gonna be related through cosine of the angle because it's the adjacent side. So my is gonna be M cosine of degrees, which is equal to 6 m multiplied by cosine of degrees. All right. Now switching over to our vector and we're gonna do the same thing. We're breaking it up. We're gonna draw the triangle down below to correspond with that angle we were given. Let me just fix that up there we go. And NX is gonna be the horizontal component down below and Y is gonna be the vertical component on the left starting with the horizontal component. OK. And this is gonna be related to um sign of the angle because it's the opposite side. So this is gonna be N multiplied by sine of degrees. N is the hypo for the magnitude of the spectrum. And so NX is gonna be equal to 5 m multiplied by sine of 40 degrees. And now for Ny and for NY, we have to be a little bit careful. And when we look at Ny actually pointing in the negative Y direction and all of our other vectors, components were pointing in the positive respective directions. OK? And Y pointed up and X pointed to the right but Y is pointing down. OK. So this is gonna be a negative value. So we have negative and multiplied by cosine of 40 degrees. It's related through cosine because of the adjacent side. So this gives us negative 5 m multiplied by cosine of 40 degrees. All right. So always when you're adding vectors, when you're breaking up vectors into components, always pay attention to what the sign is. What direction is that vector pointing and what direction is that component point? Now we have all four of our components, two vectors, two components each, let's find P now and let's start with the X component of P. OK. We've already mentioned that we can add a vector by adding the corresponding components. So let's do a component by component. The X component of P is gonna be equal to the X component of M. What's the X component of N? What are those values? Well, the X component of M is 6 m multiplied by sine of 30 degrees. And for M it's 5 m multiplied by sine of 40 degrees. So PX is gonna be equal to 6 m multiplied by sine of 30 degrees. What 5 m multiplied by sine of 40 degrees. And if we work this out on our calculator simplifying, now we get that this is approximately equal to 6. m. OK? So this is our X component and recall the X component is equivalent to the I component and those two represent the same thing. So our, our component is gonna be 6.21 m looking at our answer traces A B and D all have 6. m I, so those could all be correct, but we can eliminate option C it has 9.03 m here. We know that that's not correct. Now, moving to our Y component for P doing the same thing, you know that Py is gonna be equal to my plus NY going back up, I'm just taking a look at these values. My is 6 m multiplied by cosine of 30 degrees and NY is negative 5 m multiplied by cosine of degrees. So py is gonna be equal to 6 m multiplied by cosine of 30 degrees minus 5 m multiplied by cosine of 40 degrees. We had a plus, we had a positive but we had our negative 5 m. So that positive became a negative. I working this out again, this is gonna give us a value for py of about 1.37 m. So now we have our X and Y or IMJ respectively components of our vector P and we can figure out which answer choice is correct. In the end, we can eliminate option E or a sorry. It had negative 1.37 m J OK. So that's not correct. We found that it was positive. Option B has 9.02 for the J component also incorrect. But option D has the correct value for the vector we found and, and that vector is 6.21 m I plus 1.37 m J. Thanks everyone for watching. I hope this video helped see you in the next one.